Field of the Various Embodiments
The various embodiments generally relate to systems analysis and, more specifically, to a technique for determining linear responses of nonlinear systems.
Description of the Related Art
One objective of systems analysis is to identify an input/output relationship for a linear or non-linear system. In a linear system, the output is directly proportional to the input multiplied by a fixed gain factor plus some fixed offset. In a nonlinear system, the output to input relationship may be expressed as a polynomial that includes higher powers of the input, each multiplied by coefficients that may vary, or even be zero. Many nonlinear systems exist, and physical devices typically have nonlinear characteristics beyond a limited range of linear operation. Meteorological systems, communications channels, and audio loudspeakers are examples, without limitation, of well-known nonlinear systems.
Analysis of a nonlinear system typically involves constructing a linear model of the system and then determining the linear characteristics of the model to assess quality or performance. Similarly, synthesis of a nonlinear system of typically involves constructing a linear model. Both analyses and syntheses of nonlinear systems utilize mathematical techniques to construct linear approximations to the nonlinear system.
Fourier analysis is a well-known mathematical systems analysis technique that involves decomposing a signal applied as an input to a nonlinear system into a series of sine waves. The series of sine waves includes a fundamental, or first order, component, and other higher order harmonics. A nonlinear system responds to the higher order harmonics with a different gain factor than the fundamental component and with a different gain factor than lower order harmonics. Thus, through the different gain factor, the nonlinear system generates an output that includes nonlinear distortions of the input signal.
Analyzing the quality or performance of a nonlinear system typically includes determining the linear characteristics of the system. In that regard, when synthesizing systems that are inherently nonlinear, a primary objective is to measure a result that is either linear or as close to linear as possible. Measurement of the linear response for a nonlinear system typically entails sending an extremely low level test signal through the nonlinear device under test.
Another technique to determine the linearity of a nonlinear system includes attenuating the input signal by a factor large enough to ensure that the signal always remains in the limited linear region of operation. The resulting output is then scaled up by the inverse of the attenuation factor to yield an input-output relationship that represents the linear response of the system. This approach simply avoids the nonlinear characteristics of the system and, thus, can give an unrealistic or substantially limited representation of the nonlinear system.
As the foregoing illustrates, what would be useful is a more effective technique for analyzing the linear response of nonlinear systems across a broader range of operating conditions.